The most general action for scalar-tensor theories of gravity in the metric approach can be written as
(see e.g. [1]}, [2]} for the same convention)In fact, as shown recently [1]}, such metric STT allow to describe as well Palatini and hybrid metric-Palatini ST gravities.:
\(S[g_{\mu \nu },\Phi ,\chi ] = \frac{1}{2\kappa ^2}\int _\Omega d^4x\sqrt{-g} \left[{\mathcal {A}}(\Phi )R -{\mathcal {B}}(\Phi )g^{\mu \nu }\partial _\mu \Phi \partial _\nu \Phi -\mathcal {V}(\Phi )\right] + S_\text{matter}\left[e^{2\alpha (\Phi )}g_{\mu \nu }, \chi \right]\,,\) 
